💡 when we say we want to “control for” a variable, what we’d ideally like is to compare subjects who are exactly the same in that variable.
😦 But we often can’t get that data easily
⬇️ All the methods we’re discussing today allow us to get as closer to that ideal when it’s not possible. For each method, ask yourself “how does this make the groups I’m comparing more similar?”
💡we’ve re-weighted each sample so that men comprise 80% of the sample!
Matching
Originally \((60 + 12)/100 = 0.72\) of people in the training group get jobs, and \((350 + 275)/1000 = 0.625\) of people in the control group get jobs.
Now, \(\frac{(1 \times60 + 1\times12)}{(1 \times80 + 1\times 20)} = 0.72\) of people in the training group got jobs and \(\frac{(0.16 \times 350 + 0.04 \times 275)}{0.16 \times500 + 0.04 \times 500} = 0.67\) of people in the control group got jobs.
There’s less of an impact once we “control for” the backdoor path of gender.
Matching
select matches vs weighted sample
We could also select members of the control group who match each member of the treatment group: 👯
Matching
Since A, B, and C are confounders of the effect of treat on outcome, we want to get groups that match on A, B, and C.
Matching
Distance Matching: observations are similar if they are “similar” in A, B, and C
Propensity Score Matching: observations are similar if they have a “similar” probability of being treated
Propensity Scores
In non experiments, confounders can influence both exposure and outcome
But what if we could adjust for your probability of being treated?
Propensity Scores
propensity score: probability of being exposed given some set of covariates \(\mathbf{c}\)
Rosenbaum and Rubin (1983) suggest that controlling for propensity scores can give you an unbiased estimate of the effect of exposure on outcome if:
There are no unmeasured confounders
Every subject has a non-zero probability of getting exposed
In this example, \(Z\) impacts the probability that you’re treated (\(P(treat | z = 1)\) = 50% vs \(P(treat | z = 0)\) = 75%). from:https://www.youtube.com/watch?v=PfLYPt9ur4g
where \(A\) is the treatment (\(A = 0\) did not quit smoking, \(A=1\) did quit smoking). For people who did quit, the weights are \(w_{A = 1} = \frac{1}{p(A = 1 | \mathbf{c})}\) and for people who did not quit, the weights are \(w_{A = 0} = \frac{1}{p(A = 0 | \mathbf{c})}\).
Note: the more unlikely it is that someone should be in their treatment group, the higher the weight that subject has.
\[
\underbrace{\left(After_c - Before_c\right)}_\text{diff for control} - \underbrace{\left(After_t - Before_t\right)}_\text{diff for treatment} = DiD
\]
\[
\underbrace{\left(146.4 - 134.9\right)}_\text{diff for control} - \underbrace{\left(130.1 - 84.9\right)}_\text{diff for treatment} = \underbrace{-33.7}_\text{diff-in-diff}
\]